Testing quantum expanders is co-QMA-complete

TL;DR

Estimating the mixing time of quantum expanders is computationally hard, specifically co-QMA-complete, impacting quantum computing tasks like testing constructions and understanding thermalization.

Contribution

This paper proves that estimating the spectral gap of quantum expanders is co-QMA-complete, establishing its computational complexity.

Findings

Estimating the spectral gap is co-QMA-complete.

Implications for testing quantum expander constructions.

Relevance to thermalization in open quantum systems.

Abstract

A quantum expander is a unital quantum channel that is rapidly mixing, has only a few Kraus operators, and can be implemented efficiently on a quantum computer. We consider the problem of estimating the mixing time (i.e., the spectral gap) of a quantum expander. We show that this problem is co-QMA-complete. This has applications to testing randomized constructions of quantum expanders, and studying thermalization of open quantum systems.